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Galois Theory and Theaters of Action in a Topos

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Galois Theory and Theaters of Action in a Topos View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 18 (1980) 149-164 0 North-Holland Publishing Company GALOIS THEORY AND THEATERS OF ACTION IN A TOPOS John F. KENNISON* Clark University. Worcester, MA 01610, USA Communicated by M. Barr Received 8 October 1978 0. Introduction Constructing a Galois theory for geometric fields in a topos presents certain difficulties. If K E F is a Galoisian field extension in a topos 5Z,then one can construct the internal group of K-automorphisms of F but it might be trivial even when K f F (see Example 6.6). In the topos of presheaves over a category C, a Galois group appears to be a presheaf over Cop. There are similar difficulties with splitting a polynomial over a field K in a topos. If we are splitting x2 + 1 over K in the topos of Sets, then we usually use a dichotomous procedure: if x2+ 1 is irreducible we construct K[i], if .r* + 1 is reducible we use K. This procedure is problematic for topoi since irreducibility is not a coherent condition (i.e., not geometric in the sense of [3]). A procedure for splitting polynomials is given in Section 3. See Example 6.3 also. Of course the field containing the added roots lives in a new topos, and this is an instance of the Cole Spectrum (see [l, 3,7]). 1. Theaters of action A group in a reasonably complete topos 8 can be regarded as a left exact functor C + 8 where C is the dual of the category of finitely presented groups (see [2]. Note that “left exact” means finite limit preserving.) A Galois group should really be regarded as a profinite group by which we mean a left exact functor Go + 8 where Go is the category of finite groups. We could construct a Galois theory based on profinite groups but there would be complications concerning the definition of “closed subgroup.” [This is discussed in Section 5 and related to a curious open question of whether or under what conditions “co-equivalence relations” are “co-effective” for profinite groups.] It is more convenient to base our Galois theory on the concept of a * The author is pleased to thank Sussex University for providing a stimulating place to work on this material during his recent Sabbatical leave. The author also thanks the National Science Foundation for support under NSF Grant MCS77-03482AOl. 149 150 J.F. Kennison rhea&r which incorporates the notion of a profinite group together with an object (or “theater of action”) on which the group acts. Thus a field extension will be an example of a theater since (in Sets) the profinite Galois group acts on it. Observe that when we split x2 + 1 over the reals to get the complex field, there is no intrinsic way to distinguish i from -i. The Galois group serves to measure this “non-dis- tinguishability.” The concept of a theater is based on the idea of axiomatizing the positive notion of “distinguishability.” To illustrate the definition below, imagine that we have a Galois group acting on a field and define d(x, y) (x is “distinguished from” y) to mean that no member g of the Galois group sends x to y. Further define d,(,?, y’), where x’, y are n-tuples, to mean that there is no g for which g(f) = 7. Also define the “orbit control relation” O,,(x, g), for 6 an n-tuple, to be true when the orbit of x under the Galois group is contained in (61,. , b,). Notation. 3 shall denote an n-tuple (for some n) of variables (xi, . , x,,). If u is in S, (the symmetric group on {1,2, . , n}), then Zmis the n-tuple (x,,,,, . , x,(,,). Given f and Q, then i * a shall be the (n + l)-tuple (xi, . ,x,, a). Similarly, if i is an n-tuple and y is a k-tuple, then x’ * 9 shall denote the obvious (n + k)-tuple. If f is a function (symbol), then f(Z) shall be the n-tuple (f(xl), . , f(x,)). Definition. A theater in a topos $ is an object A together with (for each n) relations d, (of arity 2n) and 0, (of arity n + 1) such that the following conditions (called axioms) hold: Axiom A (pre-apartness). (Al) Not d,,(f, 2). (A2) d,(f, 1) iff d,(y, x‘). (A3) d,(f, 2) implies d,(K 1) or d,,(j, 2). (Comment. These axioms imply that for each n the relation ‘*Not d,” is an equivalence relation on A”. There is actually a countable infinitude of axioms given by the above axiom schemes). Axiom B. (Bl) (For each u in S,) d,(f, j? iff d,L Yc). (B2) d,(.F, 1) implies d,+l(.f * a, 7 * b). [So by induction we can also infer d,+k(i * d, Y * b).] (B3) d,(x, Y) iff d,+lLf * xn, Y * ~“1. Axiom C. If x1 =x2, then d2(xl, x2, yl, yz) or YI = YZ. (Comment. See the “Remark” following Proposition 1.4, below.) Axiom D. “Every x has an orbit.” That is: V 5,. , b,, Onk 6). n (This is an infinite disjunction; the axiom entails the assumption that the relevant sup exists.) Axiom E. O,(a, 6) and /jid,+*(f * a, y * bi) imply d,(f, >‘). Gal& theory and theaters of action in a topos 151 Axiom F. (Fl) O,,(X, &) implies O,,,(X, E) if the “set of 6’s is contained in the set of c’s.” This could be formalized by the analogues of Bl, B2, B3. (F2) Oncl(x, 6 * a) and dl(x, a) imply O”(x, 6). Definition. Sometimes we omit the subscript n and write d(Z, 7) or 0(x, &), we further say “(d, 0) is theater structure on A,” etc. We define (A, d, 0) to be a prefheafer if the first five axioms (A-E) are satisfied. If (A, d, U) is a pre-theater then there is a unique way to define 0 so that (A, d, 0) is a theater. Uniqueness is proven below (1.3), the existence is obtained by 0(x, 6:, iff 3? and (T with V(x, (6 * ?),) and Ai [d(x, ci) or Vi C; = bi]. 1.1 Proposition. In the topos of Sets, (A, d, 0) is a theater precisely when there is a profinite group G acting continuously on the (discrete) Set A such that d(T, jj) ifi g(2) f y for all g E G and O(a, i) iff for every g E G there exists an i with g(u) = bi. Moreover, if we assume that G acts effectively on A (meaning that g(a) = a for all a implies g is the group identity), then G and itsprofinite topology are determined to within isomorphism. Proof. If the profinite group G acts continuously on the set A, then it is clear that axioms A-F are satisfied when d, 0 are interpreted as indicated. Conversely, let (A, d, 0) be a theater in Sets. Define gc A XA to be pre- admissible if d(f, y) implies that (xi, y,)& g for at least one i. We say that g is admissible if it is maximally pre-admissible. Every pre-admissible relation is a function from a subset of A into A in view of axiom C. We claim that if g is admissible then the domain of g is all of A. Assume that a in A is not in the domain of g. Then for each b in A there must exist (2, p) with d(Y * a, y* 6) and g(Z) = 7, otherwise g u {(a, b)} is pre-admissible. Choose b so that O(a, 6), then one can readily find (2, y) with d(i * a, 7 * bj) for each j and g(Z) = p. By axiom E we obtain a contradic- tion, proving the claim. Observe that g is pre-admissible iff the inverse relation, g-’ is pre-admissible. The same is true of admissible relations so the above arguments show that if g is admissible then g is one-to-one and onto. The identity function is admissible by (Al) and a composition of admissible functions is admissible by (A3). It is easily shown that d (i, jj) iff g(Z) # J for all admissible g. (Since d (a, ~7)fails iff {(xi, yi)} is pre-admissible.) We also claim that O(a, &) iff the orbit of a is contained in PI,. , b,}. For if g(a) = c and if c # bi for all i, then d(a, a, c, bi) and hence d(a, c) by axioms C and E, contradicting g(a) = c. Conversely, if O(a, 6) we can then eliminate those bi for which d(a, bi), by (Fl) and (F2). Thus we can assume that O(a, &) where each bi is in the orbit. Hence if 5 contains the entire orbit we have O(a, E) by (Fl). In order for G to act continuously on A its topology must be at least as big as the topology of pointwise convergence with basic open sets W(d, 6) + {g/g(&) = b). Since 152 J.F. Kennison the orbits are finite the topology of pointwise convergence can be shown to be compact (and Hausdorff) so there is no strictly larger compact topology.
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